It is straightforward to show that Nash equilibria exist in games with continuous utility and countable players: Fudenberg and Levine prove this by taking the standard Nash existence proof with finite players, constructing a sequence of such mixed strategies with more and more players, showing that this sequence has a limit by the Bolzano-Weierstrass property for compact metric spaces (of which the mixed extension is an example), and using continuity to show that limit is an equilibrium. Existence is less obvious when there are an uncountable continuum of agents, however. This new paper by Salonen extends earlier results by Schmeidler. The proof has many technical details, but the critical part is to note that since utility is continuous on a compact metric space, the utility can only depend on countably many opponent strategies. We can then use well-ordering and transfinite induction on this countable set.
This paper is a bit too far toward the “mathematics for its own sake” edge of economics for my liking, but properties of Nash when players are infinite are actually rather important. Ehud Kalai has a 2004 Econometrica where he shows that as the number of players grows large, the equilibria of a game become robust to many things, such as private information, the sequence of moves, etc. I am doing work on this area – what might be called “robust games” – because while useful refinements of Nash have more or less been exhausted, equilibria that are robust to model misspecification on the part of the researcher have not been so exhausted; later this summer, I will discuss these problems here at greater length.
http://www.springerlink.com/index/QP7854PG82154512.pdf (LINK IS GATED. I am unable to find an ungated version of this paper, which was recently published in IJGT)
This paper does not generalize previous results of Schmeidler and the author is quite explicit about this. The formulation is a bit odd for applications, because the product sigma-algebra is too sparse to contain most interesting events. If every player has two actions $A$ and $B$ and the player set is the unit interval, then events such as “At least half of the players play $A$.” are not measurable. This is related to the problem of formulating a law of large numbers with a continuum of random variables, that was raised (in economics) by Judd in a 1985 JET paper.